Question

You have a special light bulb with a \(very\) delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you can run through this bulb?

Short answer

The largest RMS current is approximately 1.06 A.

Step-by-step solution

Understanding RMS Current

The root-mean-square (RMS) current, denoted as \( I_{rms} \), is a measure of the effective value of an alternating current (AC). It is related to the peak current \( I_{peak} \), which in this case is the maximum allowable current of 1.50 A.

Relationship Between RMS and Peak Current

For an AC current, the relationship between the RMS current and the peak current is given by the formula: \[ I_{rms} = \frac{I_{peak}}{\sqrt{2}} \] This formula assumes a sinusoidal AC waveform.

Substitute the Known Values

We substitute the given peak current into the formula: \[ I_{rms} = \frac{1.50 \, \text{A}}{\sqrt{2}} \] This calculates the maximum RMS current that does not exceed the peak current, thus preventing the wire from breaking.

Calculate Using Square Root

Calculate the division: \[ I_{rms} = \frac{1.50 \text{ A}}{1.414} \approx 1.06 \text{ A} \] Here, 1.414 is an approximation of \( \sqrt{2} \).

Key concepts

AC current

Alternating Current, or AC, is a type of electrical current in which the flow of electric charge periodically reverses direction. In contrast to Direct Current (DC) where the electric charge flows in a single direction, AC current flows back and forth. This type of current is commonly used in households and industries, primarily because it is easy to transform into different voltages using a transformer.
AC current is advantageous for the transmission of electricity over long distances. This is because it can easily be converted from high to low voltages and vice versa. The ability to control the voltage means that AC can be transmitted more efficiently over long distances compared to DC. Here's what makes AC current unique and beneficial:

• Efficient for power distribution over long distances.
• Can easily transform voltage levels with transformers.
• Widely used in domestic and commercial power supply networks.

Understanding AC current is fundamental, especially when learning about how household appliances and electrical systems work safely and efficiently.

peak current

Peak current refers to the maximum instantaneous current value reached by an alternating current within a cycle. It's crucial to understand peak current when dealing with electronic devices because it represents the highest current that would flow through the circuit, possibly causing damage if exceeded.
The peak current is significant when designing or using electrical equipment such as the delicate wire filament in a light bulb. Knowing the peak current helps determine the limits to keep the equipment safe. For example, if the peak current exceeds a light bulb's maximum tolerance, the filament risks breaking. Key points about peak current:

• Maximum instantaneous current.
• Critical in preventing damage to electrical components.
• Forms the basis for calculating the root-mean-square (RMS) current in AC systems.

By managing peak current properly, we ensure the longevity and safety of electrical appliances and systems.

sinusoidal waveform

The sinusoidal waveform is one of the most common waveforms encountered in AC electricity. It describes how the voltage and current vary periodically, following a smooth repetitive oscillation similar to a sine wave. This wave form is symmetrical and continuous, providing a consistent model for AC signals.
The importance of sinusoidal waveforms in AC systems lies in their ability to produce efficient power transfer. Sinusoidal waveforms are not just easier to analyze mathematically; they also reduce the amount of electrical noise, making them ideal for a wide range of applications. Some properties of sinusoidal waveforms include:

• Symmetrical oscillation resembling a sine wave.
• Efficient energy transfer with minimal power loss.
• Ability to be expressed mathematically, simplifying electricity calculations.

In electronics, using sinusoidal waveforms ensures improved performance and reduced energy wastage, making them essential in power and signal processing.

electrical safety

Electrical safety encompasses protocols designed to protect users from hazards like electric shock, burns, or fire emanating from electrical appliances. It's crucial when dealing with alternating current because improper handling can result in serious injuries or damage. Understanding and implementing electrical safety measures is foundational in maintaining a secure environment when operating any electrical system.

When working with AC systems, several safety practices should be followed:

• Never exceed the maximum ratings of electrical devices such as the peak current.
• Regularly inspect electrical appliances for wear and tear.
• Ensure proper grounding to prevent electric shocks.
• Use protective devices like fuses and circuit breakers.

By adhering to these safety measures, the risks associated with electrical systems can be significantly reduced for both individuals and equipment. It ensures not only personal safety but also protects assets and maintains the reliability of the equipment over time.